本文 総合研究大学院大学学術情報リポジトリ 甲1219 本文

Compounds on the 1 nm Scale:

Geometric, Electronic, and Optical

Properties

Takeshi Iwasa

DOCTOR OF

PHILOSOPHY

Department of Structural Molecular Science

School of Physical Sciences

The Graduate University for Advanced Studies

2008

Contents i

1 Introduction 7

1.1 Thesis overview . . . 7

1.2 Nanocluster science . . . 7

1.2.1 Bare metal clusters . . . 8

1.2.2 Gold cluster compounds . . . 8

1.2.3 Ordered nanoparticle assembly . . . 9

1.3 Optical response of nanoclusters . . . 10

1.3.1 Toward dynamical properties . . . 10

1.3.2 Optical response in 1-nm-sized nanoclusters . . . 11

2 Gold cluster compounds 15 2.1 Gold-thiolate cluster: Au25(SCH3)18 . . . 16

2.1.1 Computational details . . . 16

2.1.2 Geometrical structures . . . 18

2.1.3 Electronic properties . . . 25

2.1.4 Short summary . . . 31

2.2 Au13 oligomeric clusters . . . 33

2.2.1 Method of computation . . . 33

2.2.2 Geometrical structures . . . 34

2.2.3 Optical properties . . . 35

2.2.4 Short summary . . . 39

3 Development of optical response theory 41 3.1 Problems to be addressed . . . 41

3.2 Multipolar Hamiltonian . . . 42

3.3 A molecule interacting with a near-field . . . 43

3.3.1 Nonuniform light-matter interaction model . . . 44

3.3.2 Near-field radiated from an oscillating dipole . . . 46

3.4 Light-matter interaction in the Kohn-Sham equation . . . 47 i

4 Computational applications and system details 49

4.1 Time-dependent Kohn-Sham approach in real space . . . 49

4.2 Molecular system and computational details . . . 50

5 Nonuniform electronic excitation induced by the near-field 53 5.1 Nonuniform electronic excitation . . . 53

5.2 Even and Odd Harmonics . . . 57

5.3 Control of harmonic generation . . . 59

6 Conclusion 61 Appendices 63 A Quantum electrodynamics 65 A.1 Longitudinal and Transverse vector fields . . . 65

A.2 Minimal coupling Hamiltonian in the Coulomb gauge . . . 67

A.2.1 Density and polarization operators . . . 68

A.2.2 Intermolecular Coulomb interaction . . . 69

A.3 Canonical transformation and multipolar Hamiltonian . . . 70

A.4 Semiclassical equation of motion . . . 74

B About magnetic interactions 77

Bibliography 81

The aim of this thesis is to theoretically study geometric, electronic, and optical prop- erties of one-nanometer sized cluster compounds. The thesis is composed of two parts. In the first part, the geometric and electronic properties of gold-thiolate cluster com- pounds, which have recently been studied experimentally, are revealed. I will discuss how the local geometric structures are related to the electronic properties of the com- pounds. In the second part, optical response theory that is applicable to the nan- ocluster compounds is developed. Special emphasis is placed on nonuniform electronic excitations induced by near-fields.

Let me briefly review history of metal nanoclusters. Research in nanocluster com- pounds has its root on the study of bare metal clusters in gaseous phase, where size- dependent physicochemical properties are the main concern. However, most of these bare clusters are energetically and chemically unstable. In the past few decades, metal clusters protected by organic molecules have been synthesized in solution, and some of these cluster compounds were found to be stable even in the air. Although these nanocluster compounds were expected to be promising candidates for functional nanomaterials in a wide range of nanotechnologies, it is not trivial to characterize their detailed structures. Reducing the size of clusters to the 1 nm scale, their geometries and other properties become much more sensitive to the change in size and chemical compositions. In such circumstances, sub-nanometer sized gold-cluster compounds have intensively been synthesized with the definitive determination on the chemical compositions. Despite the brilliant results, even their geometrical structures have not sufficiently been characterized. Furthermore, the studies on their optical properties are still in the juvenile stage. For these reasons, I theoretically study the geometric, electronic, and optical properties of some representative cluster compounds at the 1 nm scale.

The geometric and electronic structures of a gold-methanethiolate [Au25(SCH3)18]⁺ are investigated by carrying out the density functional theory (DFT) calculations. The obtained optimized structure consists of a planar Au₇ core cluster and Au-S com- plexes, where the Au7plane is enclosed by a Au12(SCH3)12ring and sandwiched by two Au₃(SCH₃)₃ ring clusters. This geometry differs in shape and bonding from a gener- ally accepted geometrical motif of gold-thiolate clusters that a spherical gold cluster is

1

superficially ligated by thiolate molecules. This newly optimized gold-methanthiolate cluster shows a large HOMO-LUMO gap, and calculated X-ray diffraction and absorp- tion spectra successfully reproduce the experimental results. On another gold cluster compound [Au₂₅(PH₃)₁₀(SCH₃)₅Cl₂]²⁺, which consists of two icosahedral Au₁₃ clus- ters bridged by methanethiolates sharing a vertex gold atom and terminated by chlo- rine atoms, the DFT calculation provides very close structure to the experimentally obtained gold cluster [Au₂₅(PPh₃)₁₀(SC₂H₅)₅Cl₂]²⁺. I further demonstrate that a vertex-sharing triicosahedral gold cluster [Au₃₇(PH₃)₁₀(SCH₃)₁₀Cl₂]⁺is also achieved by bridging the core Au13 units with the methanethiolates. A comparison between the absorption spectra of the bi- and triicosahedral clusters shows that the new elec- tronic levels due to each oligomeric structure appear sequentially, whereas other elec- tronic properties remain almost unchanged compared to the individual icosahedral Au₁₃cluster. These theoretical studies have elucidated the fundamental properties of the promising building blocks such as geometric structures and stability of real cluster compounds in terms of the detailed electronic structures. As a next step, I have to gain a further insight into the dynamical optical properties of cluster compounds. In partic- ular for discussing photoinduced dynamics in nanoclusters or nanocluster assemblies, inter-cluster near-field interactions should be understood properly. The conventional light-matter interaction based on available lasers is quite different from the near-field interaction. The electric fields of available lasers usually have the wavelength much longer than the size of the local structure of the cluster compounds. In other words, the 1-nm-sized cluster compounds feel the almost uniform electromagnetic field and thus the local structures of the compounds cannot be resolved. In contrast, a near-field interaction occurs at the same scale of the cluster compounds and is thus expected to be used to observe the local structure of the 1 nm sized materials. The difficulty in their theoretical description arises from the fact that the near-field has a non-uniform local structure. For these reasons, I will develop an optical response theory that is applicable to 1-nm-sized clusters interacting with the near-field.

The optical response theory is developed in a general form on the basis of the multipolar Hamiltonian derived from the minimal coupling Hamiltonian by a canon- ical transformation. The light-matter interaction in the multipolar Hamiltonian is described in terms of the space integral of inner product of polarization and electric field, whereas the minimal coupling Hamiltonian uses momentum and vector poten- tial, which are rather inconvenient for practical computations. Noteworthy is the fact that the polarization in the integral can be treated entirely without any approxima- tions. This means an infinite order of multipole moments is taken into account. Thus the present approach is a generalization of the optical response formulation beyond the dipole approximation. I have incorporated the optical response theory with the nonuniform light-matter interaction into an electron-dynamics simulation approach based on the time-dependent density functional theory (TDDFT) in real space. To elucidate the electron dynamics of 1 nm-sized molecules induced by the nonuniform light-matter interaction, the integrated TDDFT approach has been applied to and

computationally solved for a test molecular system, NC6N, in the dipole radiation field. Several unprecedented electronic excitation modes were induced owing to the nonuniform light-matter interaction using the near-field in contrast to the uniform light-matter interaction that corresponds to the conventional dipole approximation. For example, high harmonics were generated more easily. It has also been found that the near-field with different phase and spatial structure promotes or suppresses high harmonics.

In conclusion, I have revealed the geometric and electronic properties of gold- thiolate nanocluster compounds and developed optical response theory in an effort to understand nonuniform light-matter interaction between near-filed and 1nm-sized cluster compounds.

I would like to acknowledge and extend my heartfelt gratitude to the following people who have made the completion of this dissertation possible:

First of all, I am extremely thankful to Prof. Katsuyuki Nobusada. His valuable and consistent encouragement and support have propelled me to start and eventually complete this dissertation.

My sincerest gratitude also goes to Prof. Tatsuya Tsukuda and Prof. Yuichi Negishi, who have furnished me with their important data and participated in fruitful discussions.

Also, I would like to thank Prof. Tomokazu Yasuike for the assistance he extended and the inspiration he provided. Dr. Masashi Noda has made considerable improve- ments in our TDDFT program, which helped me so much. My thank-you list also includes all laboratory members who have helped me throughout this dissertation. Among them, Dr. Kazuya Shiratori, who garnered his Ph.D. last year, supported me with daily discussions for this study.

To the members of other groups; Prof. Yasufumi Yamashita, Dr. Nobuya Maeshima, Dr. Satoshi Miyashita, Dr. Yasuhiro Tanaka, Mr. Wataru Mizukami: thank you for helping me out with quantum and/or computational physics.

This list is, of course, topped by the most important people in my life—my family and friends.

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Introduction

1.1 Thesis overview

This thesis is composed of two parts. In the first part (chapter 2), theoretical studies of the static physicochemical properties of gold cluster compounds, such as geometric and electronic structures and photoabsorption spectra are presented. The choice of the gold cluster compounds was due to a series of recent seminal experiments that succeeded for the first time in synthesizing and isolating gold thiolate clusters at the 1 nm scale with high precision [1–6]. The detailed static properties of some gold cluster compounds are fully discussed by resorting to the density functional calculations.

In the second part, I theoretically study dynamical optical properties of nanoclus- ters at the 1 nm scale. Nanoclusters have received much attention due to the potential applications to molecular-sized quantum devices associated with their unusual physic- ochemical properties of electronic structures, optical response, magnetism, catalysis and so forth [7–11]. Among those fascinating properties, this thesis is mostly motivated by recent appealing experiments that had demonstrated surface plasmon propagation through an array of nanoparticles [12–22]. The surface plasmon is strongly related to near-fields, which are observed only around nanostructures [23–44]. Optical near- field interactions between nanostructures have been studied in various ways [45–61]. Nevertheless, a theory of interaction between the near-field and the 1-nm-sized mate- rials, particularly in the time domain, has not yet been well established. It is, thus, highly desired to develop such an optical response theory. In chapters 3-5, the opti- cal response theory for 1-nm-sized nanoclusters is developed with special emphasis on understanding a nonuniform light-matter interaction induced by a near-field.

1.2 Nanocluster science

This section explains the interesting and important properties of gold cluster com- pounds, starting with a brief historical overview.

7

1.2.1 Bare metal clusters

Research in nanocluster compounds has its root on the studies of bare metal clus- ters in gaseous phase, where size-dependent physicochemical properties are the main concern [62,63]. While most of bare metal clusters are energetically and chemically un- stable, some clusters with specific size have been found to be unusually stable. These specific clusters are called magic clusters. Some representative magic gold clusters of Au₁₃, Au₅₅, · · · have been intensively studied [64–69]. Despite their unusual size- distribution, it is difficult to prepare size-selective or monodispersed magic clusters. The magic clusters, even if they are prepared size-selectively, are in general unstable in the air with room temperature. Furthermore, the bare metal clusters usually have simple structures close to a spherical symmetry. For these reasons, the bare metal clus- ters have so far been mainly investigated in the context of basic cluster science and have not been discussed with the aim of developing cluster-assembled materials. Such conventional cluster science turned around after the method to prepare and isolate metal clusters size-selectively by protecting organic molecules had been established. In the next section, I will explain the importance of metal-molecule, in particular gold-thiolate, cluster compounds.

1.2.2 Gold cluster compounds

Small gold clusters protected by thiolate molecules have received much attention due to their unique physicochemical properties such as optical response, catalysis, and magnetism [70–74]. The thiolated gold clusters are also of increasing importance in the rapidly growing area of nanotechnology because they are expected to be one of the prototypes of molecular-sized materials which might function as optical devices, electrical junctions, and chemical sensors [75–81].

For example, some groups observed giant magnetic moments in thiolated gold surfaces [82] or thin films [83, 84]. The key ingredient to understand the giant mag- netic moments is localized electrons transferred from the gold substrate to the thiolate molecule. Vager and Naaman proposed a theoretical model that the transferred elec- trons form electron pairs in the triplet state (i.e., boson electron pairs) and these boson pairs induce the giant magnetic moments [85]. On the other hand, Hernando et al. explained the unexpected magnetism as due to the blocking of a local magnetic mo- ment by the giant magnetic anisotropy [83, 84]. They suggested that the transferred localized electrons induce such giant anisotropy through spin-orbit interaction.

Such unexpected magnetism was observed similarly in thiolated gold nanopar- ticles [72, 86–88]. Crespo et al. reported that the thiol capped gold nanoparticles, whose averaged core size is ca. 1.4 nm, showed ferromagnetic hysteresis with mag- netic moment per Au atom µ = 0.036 µ_B [72]. However, the value of the magnetic moment is several orders of magnitude smaller than those of the thiol capped gold thin films [83,84]. Furthermore, they found that the gold nanoparticles stabilized by weak- interacted ligands of tetraoctyl ammonium bromide became diamagnetic. In contrast,

Yamamoto et al. reported that the X-ray magnetic circular dichroism (XMCD) study revealed the intrinsic magnetism consisting of a superparamagnetic part obeying the Curie law and of a temperature-independent Pauli-paramagnetic part in gold nanopar- ticles embedded in weak interacted ligands, poly(N-vinyl-2-pyrrolidone) [88,89]. They attributed such magnetism to the mixture of ferromagnetism of the surface gold atoms and the Pauli-paramagnetism of the core atoms. As completely opposite to the result obtained by Crespo et al., they observed that the strong interacting ligand such as thi- olates quenched the surface ferromagnetism. The magnetism observed in various types of gold-thiolate systems, particularly in nanoparticles, is still controversial. These con- troversial issues concerning the magnetic properties in gold-thiolate systems are partly raised owing to the fact that the detailed electronic structures of these gold-thiolate systems were not fully specified. In other words, spin-polarized electronic structures of these systems have not been clarified.

Since these properties depend on their cluster sizes and structures, progress in the synthesis of gold-thiolates with well-defined chemical compositions is crucial in not only fundamental but also applied sciences. A number of methods to prepare and isolate monodispersed gold-thiolates have so far been developed [71,90–95] and in a recent pa- per Negishi et al. achieved the size-separated synthesis of glutathione (GSH)-protected gold clusters, Au-SG [1]. In the paper, they decided the definitive chemical compo- sition of a series of the clusters, Au₁₀(SG)₁₀, Au₁₅(SG)₁₅, Au₁₈(SG)₁₄, Au₂₂(SG)₁₆, Au₂₂(SG)₁₇, Au₂₅(SG)₁₈, Au₂₉(SG)₂₀, Au₃₃(SG)₂₂, and Au₃₉(SG)₂₄, with high-resolution mass spectrometry, although their geometrical structures remain unresolved. There- fore, it is highly desirable to specify the geometric structures of these cluster com- pounds. In 2.1, the theoretical investigation on the geometry of one of these Au-SG clusters will be discussed.

1.2.3 Ordered nanoparticle assembly

As mentioned above, nanometer-sized metal clusters have been under extensive in- vestigation owing to their novel physicochemical properties, which are known to be significantly different from those of the corresponding bulk metals. Metal nanoclusters are also expected to be key ingredients in new materials that function as molecular- sized quantum devices [75, 81, 96]. There is a rapidly growing understanding of fun- damental properties of each individual metal nanocluster. However, relatively little is known about whether these clusters retain their individual properties after assem- bly as well as produce new collective features due to aggregation. The most direct approach to this issue is thought to specify a unit cluster that serves as a building block of cluster-assembled compounds, and then determine how the assembled com- pounds are constructed from the units. Nevertheless, it is not trivial to construct such cluster-assembled compounds in a bottom-up approach because each metal clus- ter easily coalesces into an aggregate through altering their individual geometrical and electronic structures. In general, the original physicochemical properties become rather obscure after aggregate formation and then the properties of bulk metals are

dominant.

To realize the cluster-assembled compounds with electronic properties of each metal constituent, the ”clusters of clusters” concept proposed by Teo and cowork- ers [97–99] is suggestive. In a series of extensive investigations, they demonstrated that oligomeric metal-cluster compounds were systematically synthesized in a step- wise manner by aggregating icosahedral metal clusters. In such clusters of clus- ters, the individual icosahedral clusters serve as the basic building blocks and then form a polyicosahedral cluster through sharing vertex atoms. Khanna and Castleman and their coworkers have intensively demonstrated that aluminum-based icosahedral clusters (referred to as ”superatoms” in their studies) form cluster-assembled com- pounds [100, 101]. The key in both concepts of clusters of clusters and superatoms is that the assembled compounds are constructed from building units retaining the elec- tronic properties of the constituent units. Similarly to the polymerized clusters based on cluster of cluster, very recently a gold cluster-assembled compound was synthe- sized [6]. This cluster compound is regarded as a Au13 dimeric one in which two Au13

clusters share one vertex gold atoms. For larger assemblies, it is crucial to understand how the constituent unit clusters are assembled. In 2.2, I will explain the mechanism of oligomerization of the gold cluster compounds.

1.3 Optical response of nanoclusters

1.3.1 Toward dynamical properties

The first step to achieve cluster-based devices at the nanometer scale is to understand the static physicochemical properties of the constituent building blocks, in both iso- lated and assembled states. Once obtained these properties, the next step is to study their dynamical properties associated with electric current, energy transfer, and chem- ical reactivity. In particular, for discussing photoinduced dynamics in nanoclusters or nanocluster assemblies, we should have proper understanding of an inter-cluster near- field interaction. Several recent appealing experiments concerning near-field excitation dynamics in nanoparticle systems will be reviewed.

I would like to mention about recent appealing experiments, which motivate the second part of this thesis, of a propagation of surface plasmon or electromagnetic en- ergy in weakly-interacting ordered metal nanoparticle systems at the hundred nanome- ters [12–22]. These experiments are schematically explained in Fig.1.1. The or- ange and blue spheres represent nanoparticles and near-field interactions, respec- tively, and the black illustrates the tip of a scanning near-field optical microscopy (SNOM) [23–44, 102–110]. The leftmost particle is locally irradiated by the tip and then the electromagnetic energy transfers from the tip to the particle through the near-field interaction. The electromagnetic energy is subsequently propagated along the array also through the near-field interaction. Experiments of local excitation of sin- gle gold nanostructure using SNOM also inspired this thesis [34,37,38,40,103,111–113]. In these experiments, surface plasmon has been induced by the near-field illumination

near-field interaction

nano particle near-field

electromagnetic energy transfer propagating light

Near-field microscope

Figure 1.1: The image of photonic currents, where the orange sphere, the blue circle, and the black object, respectively represent metallic nanoparticles, near-field interac- tions, and a SNOM tip. An array of nanoparticles has been excited by a near-field around the probe tip (left) and then an electromagnetic field is propagated through the array from the left to the right.

which excites a local part of the wavefunction of the gold nanostructure. The near-field illumination of the single gold nanostructure can be considered as a local excitation of a coherent wavefunction. In contrast, the surface plasmon propagation through the nanoparticle array can be considered as a local excitation of an incoherent system. The near-field interaction between each nanoparticle introduces a coherency into the arrayed system.

Optical properties of metal nanoparticles, especially noble metal nanoparticles, have been extensively studied both experimentally and theoretically because of their ability to largely increase the measured signal of the Raman spectroscopy, which is known as Surface Enhanced Raman Spectroscopy (SERS) [114–120]. One of the origin of SERS is considered to be a largely enhanced electromagnetic field caused by a surface plasmon of a metal nanoparticle. Between closely spaced nanostruc- tures, largely enhanced electromagnetic fields have also been reported and used as a nonlinear-optical source or chemical reaction field aiming at a new type of chemical reactions [102, 121–124].

1.3.2 Optical response in 1-nm-sized nanoclusters

The above-mentioned experiments clarify the two essences of the near-field; the large enhancement of an electromagnetic field intensity and the localized character. For the first one, the intensity of near-fields is enhanced by a surface plasmon induced by an incident light. The incoming propagating light oscillates electrons in a nanostructure and this oscillation (the surface plasmon) generates a new electromagnetic field around the surface of the nanostructure. The electrons then interact with the newly generated

propagating light

scattered light

near-field

=enhanced, nonuniform electromagnetic field molecule

Figure 1.2: SERS is schematically illustrated. The orange particle and the blue circles represent a metallic nanoparticle and a near-field interaction, respectively. A molecule on the surface of the nanoparticle is exposed to the near-field and scatters the light.

field and regenerate an electromagnetic field. This feedback loop continues endlessly. Thus, theoretically, the field enhancement requires a self consistent solution for light- matter interactions. The field enhancement for metal particles at the 100 nm scale has been studied theoretically by solving the Maxwell’s equations using the Green’s function method [23–25], the Finite Difference Time-Domain (FDTD) method [125– 132], the Multiple MultiPole method [32, 133], and/or Discrete Dipole Approximation (DDA) [51]. The method of Green’s function is also used to analyze the surface plasmon in gold nanostructures induced by the near-field illumination [34, 37, 38, 40, 103,111–113]. These field enhancement are, however, not necessarily important at the 1 nm scale, because the surface plasmon owes much to the particle size and does not occur at the 1 nm scale. Therefore, we can concentrate on the nonuniform light-matter interaction to study near-field interaction of nanoclusters at the 1 nm scale. For the second property, the localized character can be understood as that a near-field has a nanoscale spatial structure. As a result, a material in the near-field will interact with a nonuniform electromagnetic field.

The quantum mechanical treatment is required for the nanoclusters at the 1 nm scale. Whereas a near-field is still able to be treated as a classical electromagnetic field as long as the field intensity is rather strong. The strong means that one or two photon absorption or emission due to a light-matter interaction does not largely decrease or increase the intensity of the field. An optical response theory based on this assumption is called the semiclassical theory, that is, the quantum mechanics is used for matter and the classical electrodynamics for light. Under this semiclassical approach, the most rigorous treatment is to solve the coupled Schr¨odinger and the Maxwell’s equations self consistently [45, 47, 54, 55, 134]. Although this theory can

be used for considering non-uniformity and field enhancement simultaneously, actual computations are computationally demanding. On the other hand, near-field inter- actions between two quantum mechanical particles have been studied [56–61], where interparticle interaction is reduced to be dipole-dipole or dipole-multipole interactions using simple metallic particles. However, the effect of the nonuniform electromagnetic field on electronic excitations was not clearly understood by these studies. Thus, I establish a suitable theory to reveal the nonuniform electronic excitation.

The theory developed in this thesis is based on a semiclassical treatment that 1- nm-sized materials are treated quantum mechanically while the electromagnetic fields are treated classically. Main focus of this study is the nonuniform electronic excitation of molecules at the 1 nm scale. Thus, the special emphasis is placed on an explicitly taking account of the spatial distribution of the electromagnetic field.

Gold cluster compounds

• ”Theoretical Investigation of Optimized Structures of Thiolated Gold Cluster [Au₂₅(SCH₃)₁₈]⁺”, T. Iwasa, and K. Nobusada, J. Phys. Chem. C, 111, 45-49, (2007)

• ”Thiolate-Induced Structural Reconstruction of Gold Clusters Probed by¹⁹⁷Au Mossbauer Spectroscopy”, K. Ikeda, Y. Kobayashi, Y. Negishi, M. Seto, T. Iwasa, K. Nobusada, T.Tsukuda, and N. Kojima J. Am. Chem. Soc.(communications), 129, 7230, (2007)

• ”Gold-thiolate core-in-cage cluster Au₂₅(SCH₃)₁₈shows localized spins in charged states”, T. Iwasa and K. Nobusada, Chem. Phys. Lett. 441, 268-272, (2007)

• ”Oligomeric Gold Clusters with Vertex-Sharing Bi- and Triicosahedral Struc- tures”, K. Nobusada, and T. Iwasa, J. Phys. Chem. C (Communication), 129, 7230, (2007)

Geometric, electronic and optical properties of the unusually stable gold-thiolate cluster Au25(SCH3)18 and the bi- and triicosahedral clusters are theoretically studied in this chapter. Ch. 2.1.2 decides the geometrical structure of Au₂₅(SCH₃)₁₈ in its cationic state (1+) with a closed electronic shell structure. In 2.1.3 and 2.1.3, I will investigate the spin polarization and absorption spectra of the gold-thiolate cluster [ Au₂₅(SCH₃)₁₈]ⁿwith different charged states n = 3−, 2−, 1−, 0, 1+, and 3+. Ch. 2.2.2 describes the geometrical and electronic structures of bi- and triicosahedral clusters and the relationship between the geometries and absorption spectra will be discussed in 2.2.3. To this end, I have performed DFT calculations of the cluster explicitly taking account of the spin multiplicity.

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2.1 Gold-thiolate cluster: Au

(SCH

)

Very recently, Negishi and Tsukuda experimentally confirmed that Au₂₅(SG)₁₈ was unusually thermodynamically [1] and chemically stable in comparison with the other Au-SG clusters [2]. The report of large-scale synthesis of Au25(SG)18 supports its extraordinarily high stability [3]. They also found that the pattern of the absorption spectrum of Au₂₅(SG)₁₈ was rather insensitive to change of the ligand (SG) for other thiolate molecules, in sharp contrast to the other Au-SG clusters that showed ligand- sensitive absorption spectra. These experimental observations imply the existence of a characteristic Au₂₅S₁₈ framework. Concerning the framework, on the basis of the X-ray diffraction (XRD) spectrum of Au₂₅(SG)₁₈, they have proposed that this cluster has a Au₂₅core cluster with a face-centered-cubic based structure. Unfortunately, the detailed geometrical structure was still unclear. Thus, to study the physicochemical properties of the Au₂₅(SG)₁₈ cluster, I begin with deciding the geometrical structure. The XMCD study of a series of the Au-SG clusters was also reported by Negishi and Tsukuda [4]. This is the first study that the magnetic properties were investigated for the fully size-specified gold-thiolate clusters. Although they found that the Au-SG clusters were spin-polarized, the clusters were only paramagnetic with µ = 0.0093µ_B per Au-S bond. On the other hand, in metal cluster compounds, it has been known that their stabilities are strongly affected by the number of electrons. A charging thus would have a nontrivial impact on its electronic properties such as absorption spec- trum. However, on Au25(SG)18, change in the charge states unaffects its absorption spectrum [5]. In 2.1.3, I will discuss the charging effects on the absorption spectrum of the gold-thiolate cluster.

2.1.1 Computational details

I have carried out DFT calculations for the gold-methanethiolate cluster [Au₂₅(SCH₃)₁₈]⁺, which is a theoretical model for the glutathione-protected gold cluster Au25(SG)18. I have adopted such a simplification of ligands frequently used in the previous calcu- lations [135–140] because the full geometry optimization of Au₂₅(SG)₁₈ requires in- credibly large computational costs. Furthermore, the above-mentioned experimental result, i.e. the absorption spectrum of Au₂₅(SL)₁₈having a lack of dependence on the ligand L, partly supports this modeling. The present model cluster was calculated in its cationic state with a closed shell structure to avoid considering explicit spin effects. Charging effects on the geometric structure are mentioned in Section 3.

The geometry optimization was starting from two types of initial structures with different Au₂₅core structures. One of the Au₂₅cores is based on a face-centered-cubic (fcc) structure with six Au(111) facets consisting of eight gold atoms as shown in Figure 2.1(a). The other one is based on a vertex-sharing centered icosahedral Au₁₃ dimer as shown in Figure 2.1(b). In this thesis, I refer to these clusters as FCC- Au₂₅ and SES-Au₂₅, respectively. FCC-Au₂₅ is chosen following the experimental result of the XRD spectrum, and SES-Au25 is taken into account by analogy with

a trimetalic cluster coordinated by triphenylphosphine (Ph3P-) and chlorine (Cl) lig- ands, [(Ph₃P)₁₀Au₁₂Ag₁₂PtCl₇]Cl [141]. The initial structures of [Au₂₅(SCH₃)₁₈]⁺ are constructed by passivating each core cluster with 18 methanethiolate molecules within D_3d molecular symmetry for FCC-Au25 and within C2v molecular symmetry for SES-Au₂₅.

Figure 2.1: Optimized structures of bare Au₂₅ clusters (a)FCC-Au₂₅ and (b) SES- Au₂₅.

Figure 2.2: Optimized geometry of the gold-methanethiolates (a) FCC1, (b) FCC2, and (c) SES, respectively.

In this chapter, all the quantum chemical calculations were carried out employ- ing the TURBOMOLE package of ab initio quantum chemistry programs [142]. The geometry optimizations based on a quasi-Newton-Raphson method were performed at the level of Kohn-Sham density functional theory (KS-DFT) employing the Becke three-parameter hybrid exchange functional with the Lee-Yang-Parr correlation func- tional (B3LYP) [143,144]. The triple-ζ valence-quality plus polarization (TZVP) basis

from the TURBOMOLE basis set library has been used in all calculations, along with a default 60-electron relativistic effective core potential (ECP) [145] for the Au atom. The absorption spectra were simulated by calculating the oscillator strength within time-dependent density functional theory (TDDFT). The present TDDFT is based on time-dependent Kohn-Sham response theory [146–149] Excited state properties were obtained from a pole analysis of frequency-dependent linear response functions.

The XRD spectra of the optimized structures were calculated by using the Debye formula [150]. The diffracted intensity I(s) as a function of the diffraction vector length s = ^{2 sin θ}_λ is given by

I(s) = XN i,j=1

cos θ

(1 + α cos²2θ)^exp µ

−^Bs

2

2

¶

f_if_j^sin(2πsrij) 2πsrij

,

where r_ij is the distance between the i-th and j-th atoms in the optimized gold- methanethiolates and (fi, fj) are the corresponding atomic scattering factors. θ is the diffraction angle, and λ is the wavelength of the incident X-ray beam. Angular de- pendent geometrical and polarization factors are expressed in the form of _{(1+α cos}^{cos θ}2

2θ)^,

where α is almost equal to unity for the unpolarized incident beam. The damping factor exp^³−^Bs₂^2´ means thermal effects. Since the charging effect of each atom was not taken into account, the scattering factor of an i-th atom is equal to its atomic num- ber. The equipment dependent parameters were set to be α = 1.01 and λ = 0.1051967 nm following the experiments by Negishi and Tsukuda [151]. On the other hand, the dumping parameter B = 0.005 nm² was chosen to reproduce the XRD spectrum of Au₂₅(SG)₁₈ according to what was discussed in ref [152].

2.1.2 Geometrical structures

I have obtained three types of optimized structures for [Au₂₅(SCH₃)₁₈]⁺. Two of these structures are derived from FCC-Au25, and the other one from SES-Au25. The three optimized structures are shown in Figure 2 and their structural parameters are summarized in Table 1. The atoms of Au, S, C and H are shown in gold, green, gray, and white, respectively. In the following discussion, the two optimized structures derived from FCC-Au₂₅ are referred to as (a) FCC1 and (b) FCC2, and the other one from SES-Au₂₅ as (c) SES. FCC1 expands isotropically whereas FCC2 expands laterally in comparison with the initial structure of FCC-Au₂₅. The structure of SES is bent compared with that of SES-Au₂₅. As will be discussed later in detail, FCC2 is the most preferred structure of [Au₂₅(SCH₃)₁₈]⁺. In the first place, I fully analyze the geometric and electronic structure of the FCC2 gold-thiolate cluster. Then, the obtained properties of FCC2 are compared with those of the other optimized gold- thiolate clusters, FCC1 and SES.

Table 2.1: The nearest neighboring interatomic distances of two bare Au25 clusters (FCC-Au₂₅ and SES-Au₂₅), and of three optimized gold-methanethiolates (FCC1, FCC2, and SES).

distance(˚A) FCC-Au₂₅ SES-Au₂₅ FCC1 FCC2 SES Au - Au 2.79 - 3.29 2.75 - 3.15 2.83 - 3.37 2.80 - 3.60 2.87 - 3.20

Au - S — — 2.39 - 2.72 2.39 - 2.43 2.38 - 2.41

S - C — — ∼ 1.85 ∼ 1.85 ∼ 1.85

C - H — — ∼ 1.09 ∼ 1.09 ∼ 1.09

Figure 2.3: The top-view (above) and side-view (below) of the FCC2 in space-filling model (left) and ball&stick model (right).

Figure 2.4: The four sub-systems of FCC2.

The best optimized geometrical structure of [Au25(SCH3)18]⁺

Figure 3 shows top and side views of FCC2 in different ways of drawing, ”space-filling” and ”ball and stick” models. FCC2 has an oblate structure and can be fractionalized into three subsystems; (i) a Au₇ core cluster, (ii) a Au₁₂(SCH₃)₁₂ ring, and (iii) two Au3(SCH3)3 rings. This classification is schematically drawn in Figure 2.4. The almost planar Au₇ core cluster is surrounded by the Au₁₂(SCH₃)₁₂ ring. Then, the structure of FCC2 is completed by capping this core and ring subsystem from both sides of the top and the bottom with the two Au₃(SCH₃)₃ rings. The structural details of each fractionalized subsystem are as follows: (i) The Au₇core cluster has a centered gold atom surrounded by a six-membered gold ring with a chair conformation. The Au-Au distance between the centered gold atom and each surrounding gold atom is 2.80 ˚A, and the distance between the nearest neighboring surrounding gold atoms is 2.87 ˚A. (ii) The Au12(SCH3)12ring consists of -Au-S- repeated bonds. The Au-S bond length is alternatively changed to either 2.39 ˚A or 2.40 ˚A. The nearest neighboring Au-Au distance is up to 3.60 ˚A. This Au-Au distance is significantly larger than that of usual bare gold clusters. (iii) The Au3(SCH3)3 ring also consists of -Au-S- repeated bonds whose bond length is 2.43 ˚A, whereas the Au-Au distance is 3.15 ˚A. As is clear from these structural analyses, the present optimized gold-thiolate cluster consists of the core gold cluster and the (Au-SCH3)ncomplex-like rings enclosing the core cluster. This feature is in sharp contrast to the widely known picture of gold-thiolate clusters that a core gold cluster is superficially protected by thiolate molecules. A similar finding was reported in a quite recent paper (ref [139]).

I continue the discussion of the stability of this optimized structure. From the vibrational analysis, it has been found that the structure has only one small imaginary frequency (= 12.22cm⁻¹). The existence of the imaginary frequency leads to the structural relaxation, which breaks the D_3d molecular symmetry. Thus, I have carried out geometry optimization again for FCC2 within Cs molecular symmetry to check a Jahn-Teller effect. As a result, the atomic rearrangements were very small, i.e., the largest one was only 0.004 ˚A. Therefore, I reasonably consider FCC2 to be acceptable for an energetically local minimum structure.

Before ending this subsection, the charging effect of [Au₂₅(SCH₃)₁₈]⁺ on the geo- metric structure should be addressed. I have also carried out the geometry optimiza- tion of FCC2 in its neutral state. The calculated result showed that the structure of only the Au₇ core was slightly changed but the other Au-SCH₃ structures were almost unchanged. This is simply because not more than one or two charge differences are negligible at least for the geometric structure. Furthermore, I have confirmed that the charging effect does not have a qualitative influence on the XRD spectra. Although the explicit treatment of the spin multiplicity is mandatory for revealing magnetism or spin-dependent energy levels and there might be higher spin states that are lower in energy than the present closed shell structure, the charged (closed-shell) calculations provide reasonable results, as far as the geometric structure is concerned.

Comparison with the other optimized structures

Figure 2.5: The structure of FCC1 and SES in ”ball & stick” model.

In this subsection, I compare the geometric and electronic structures of FCC2 with those of the other optimized structures, FCC1 and SES. The structures of FCC1 and SES are shown in Figure 5, (see also Figure 2). As was mentioned in the beginning of this section, the deformation of the Au₂₅ core structure caused by the coordination of the methanethiolates is commonly seen in FCC1 and SES. Such deformation is also found in the previous theoretical studies [137–140, 153] The gold and sulfur atoms in FCC1 and SES also form a -Au-S- repeated network.

FCC1 consists of fractionalized components of Au core and Au-S ring clusters as is similar to the structure of FCC2. However, there are two remarkable differences be- tween the FCC1 and FCC2 structures. First, in FCC1 a methanethiolate coordinates to a gold atom rather than forming a Au3(SCH3)3 ring. Second, the centered gold atom of the core cluster in FCC1 is localized markedly apart from the surrounding gold atoms by 4.60 ˚A. The result implies that the Au-Au interaction does not work any more, whereas the centered Au7 core cluster in FCC2 plays an important role in stabilizing the structure of FCC2 as discussed later.

The structure of SES is more complicated than those of FCC1 and FCC2, and has a rather different geometric feature. Although it is difficult to fractionalize the whole structure, SES also consists of Au core and Au-S complex-like ring clusters. Therefore, the three optimized structures are similar to each other in a sense that they are constructed from a gold core cluster and Au-S complex-like ring clusters enclosing the core. However, FCC2 only provides the Au-S ring cluster with the ideal ratio of Au and SCH₃ being 1:1.

Figure 2.6 shows Kohn-Sham orbital energy levels of FCC1, FCC2, and SES as well as two core clusters of FCC-Au₂₅ and SES-Au₂₅. The energies are indicated in units of eV relative to the energy of HOMO. I specify each component of the atomic

Figure 2.6: Kohn-Sham orbital energy levels of FCC-Au₂₅, FCC1, FCC2, SES, and SES-Au25. The energies are indicated in the unit of eV relative to the HOMO ener- gies. The orbitals colored in green, blue, yellow, red and light-blue represent Au(6sp), Au(5d), S(3s), S(3p), and CH₃ orbitals ,respectively. The broken line separate the occupied orbitals and the unoccupied orbitals. HOMO and LUMO are just under the line and just above the line, respectively.

orbitals contributing to the KS orbital energy levels by using different colors. The colors blue and green indicate Au 5d and 6sp orbitals, respectively, and the colors yellow, red and light blue S 3s, 3p, and methyl group orbitals, respectively.

The KS orbitals near HOMO and LUMO of the bare FCC-Au₂₅ and SES-Au₂₅ clusters are constructed mainly from the Au 6sp orbitals with small hybridizations of the Au 5d orbitals. The Au 5d band ranging from ca. −6 to ca. −2 eV is below the Au 6sp band. From the Mulliken population analysis, the central gold atoms of the bare clusters are negatively charged. The HOMO-LUMO gaps of both clusters are

∼1.2 eV.

Global features of the KS orbital energy levels of the three gold-methanethiolates can be classified into four groups from higher to lower in energy as follows: (i) the Au 6sp band, (ii) the Au (5d) - S (3p) bonding orbitals, (iii) the Au 5d band, and (iv) localized orbitals of the methanethiolate. The energy levels of HOMO, LUMO and the low-lying unoccupied orbitals belong to group (i). As is the case with the bare Au25 clusters mentioned above, the inside of the gold-methanethiolates is negatively charged. However, the negative charges in the gold-methanethiolates are smaller than those in the bare clusters. This is attributed to the charge transfer from the gold core to the thiolate molecule. This charge transfer was found to be the electronic transition from the Au(6s)-Au(6s) bonding orbital to the Au(5d6s)-S(3p) bonding orbital. The observed elongation of the Au-Au distance can be explained in terms of this charge transfer.

Table2.2shows the total energies and the HOMO-LUMO gaps of FCC1, FCC2 and SES. We can see that the total energy of FCC2 is lowest among these three energies.

Table 2.2: Total energies and HOMO-LUMO gaps of three optimized gold- methanethiolates, FCC1, FCC2 and SES.

FCC1 FCC2 SES

E^tot(eV) -306905.6 -306913.6 -306909.0

HOMO-LUMO gap(eV) 1.05 2.19 1.11

A surprisingly large HOMO-LUMO gap (2.19 eV) is found in FCC2. This energy gap is about twice as large as those of the other gold-methanethiolate clusters. From the orbital analysis, it has been found that HOMO, LUMO and LUMO+1 of FCC2 mainly consist of the atomic orbitals of the Au₇ core cluster. HOMO is composed of the 6s orbitals of the surrounding gold atoms and the 6p orbitals of the centered gold atom. These atomic orbitals interact with each other and split into bonding and antibonding orbitals. This strong interaction leads to the large HOMO-LUMO gap of FCC2. Comparison with experimental data:

XRD and absorption spectra

Figure 2.7: Experimental diffraction spectrum ( below, labelled ”EXP”) and diffrac- tion spectra calculated from optimized geometry of the gold-methanethiolates: FCC1, FCC2 and SES. The diffraction spectra in each case are displaced along the vertical axis for clarity of exposition.

Figure2.7shows the XRD spectra of the present three optimized structures in com- parison with the experimental data [151]. The global features of the spectra of FCC2 and SES are qualitatively in good agreement with the experimental data. Furthermore, the spectrum of FCC2 provides two major peaks (∼4 and ∼7 nm⁻¹) which reasonably coincide with the experimental ones (see, the vertical dotted lines). Here, I should

stress that Au25(SG)18 was highly purified, so that the experimental XRD spectrum reflects only this size of the cluster. Thus, it is not necessary to take account of the size distribution of the clusters depending on the experimental conditions. Although the qualitative agreement of the calculated FCC2 XRD spectrum with the experimen- tal one should not be taken as a final proof of the existence of the FCC2 structure, this result will be utilized when totally judging the most probable [Au₂₅(SCH₃)₁₈]⁺ structure.

Figure 2.8: Absorption spectra of FCC1, FCC2, SES, and experiment( below, labelled

”EXP”). The solid lines are obtained by convoluting each absorption peak with the Lorenz function for FCC1, FCC2 and SES. For convenience, the broken line is drawn to comparison for the absorption peaks.

Figure 2.8 shows the comparison of the calculated absorption spectra with the experimental data [151]. The solid lines are obtained by convoluting each absorption peak with the Lorentz function. The first major peaks of the spectra are indicated by arrows. As is clearly seen from the figure, the absorption spectrum of FCC2 sufficiently reproduces the experimental data. The first peak at 689 nm and the shoulder at 534 nm are assigned to the electronic transitions that occurred within the Au7 core cluster. The first peak corresponds to the electronic excitation from HOMO to LUMO, and the shoulder from HOMO to LUMO+1.

Brief summary

FCC2 was decided to be the most plausible geometry for the Au₂₅(SCH₃)₁₈ by the careful investigations on the electronic structure, total energy, and comparisons be- tween experiments. The following sections study further the electronic and optical properties of FCC2 by using unrestricted DFT calculations.

2.1.3 Electronic properties

Let me briefly review the unique geometrical structure of [Au₂₅(SCH₃)₁₈]⁺obtained in the previous section. The cluster consists of a Au7core and a (AuSCH3)12[(AuSCH3)3]2

cage structure as shown in Fig. 2.9. The Au₂₅ cluster is effectively reduced to the Au₇ core owing to a coordination of the eighteen thiolates. Then, the Au₇ core takes a nearly planar structure and the remaining outer gold atoms together with the thi- olates form a robust complex-like cage composed of one Au₁₂(SCH₃)₁₂ ring with a zig-zag (-Au-S-) framework and two planar Au₃(SCH₃)₃ rings. As will be numerically confirmed later, this structure is almost unchanged in the different charged states and these unique structural features play a very important role in realizing the spin polarization localized at the Au₇ core.

Magnetism

Figure 2.9: The optimized geometry of [Au₂₅(SCH₃)₁₈]⁺ consisting of Au₇ core and (AuSCH₃)₁₂[(AuSCH₃)₃]₂cage structure: Au (gold) and S (green). For simplicity, the methyl groups are not shown in this figure.

I started from the geometry optimization of the neutral cluster, Au₂₅(SCH₃)₁₈ by resorting to unrestricted DFT calculations. The ground state of the cluster was found to be the doublet state. Figure2.10(a) shows the density of state (DOS) for the up [red lines] and down [blue lines] electron spins. The short vertical line denotes the Fermi energy level. As shown in the figure, the KS orbital analysis indicates that the curves of DOS are classified into three regimes contributed mainly from (i) the Au(5d) orbitals, (ii) the Au(5d)-S(3p) bonding orbitals, and (iii) the Au(6s, 6p) orbitals of the Au₇core. The figure shows that the one unpaired electron (up spin in this figure) is localized at the Au₇ core, and the red and blue curves of DOS for Au(5d) and Au(5d)-S(3p) are almost symmetrical. The highest occupied molecular orbital (HOMO), containing one up-spin electron in the present unrestricted DFT calculations, of the neutral (n=0)

Au(5d)

Au(5d)-S(3p)

Au7(6s6p)

Orbital Energy

=~3.5eV

(a)

n = 0

(b)

n = 3-

(c)

n = 2-

(d)

n = 1-

(e)

n = 1+

(f)

n = 3+

Fermi Energy Level

DOS

Figure 2.10: Density of States for the up (red) and down (blue) spins of the charged clusters [Au₂₅(SCH₃)₁₈]ⁿ. For comparison purposes, the absolute values of the orbital energies are conveniently shifted in each figure. The short vertical lines denote the Fermi energy level. All the DOSs were obtained by convoluting each KS energy level (indicated by the red and blue vertical lines) with the Lorentz function.

cluster depicted in Figure2.11schematically demonstrates that the unpaired electron is indeed localized at the Au₇ core and almost no electron density exists around the Au-S cage. For these reasons, it is not expected that the magnetic moment is induced at the surface of the gold-thiolate cluster. This is a sharp contrast to the observations in gold-thiolated surfaces or thin films in which the giant magnetic moments were induced by the localized electrons transferred from the gold substrate to the sulfur atom, but is consistent with what was obtained by Yamamoto et al. If we consider the electrons confined in the Au₇ core as ”trapped electrons” proposed by Hernando et al., their explanation also seems to be consistent with the present result. Negishi et al. reported that the magnetic moments in the Au-SG clusters were induced by the localized electrons in the Au-S bonds and estimated them to be 0.0093 µ_B. Such a small value is consistent with the above calculated result that the significant magnetic properties are not induced at the surface of the small gold-thiolate cluster.

I have carried out geometry optimizations of the cluster in n = 3−, 2−, 1−, 1+ and 3+ charged states. It has been confirmed that these optimized structures are very similar to that of the neutral cluster except that the structure of the Au₇ core slightly winds depending on the charged states. This is because the Au-S cage enclosing the Au7 core forms a rather rigid framework as described in the previous section. The HOMO-LUMO (lowest unoccupied molecular orbital) gaps and the spin multiplicities of all the charged gold-thiolate clusters are summarized in Table 1. It should be noted that these HOMO-LUMO gaps do not necessarily correspond to an absorption edge because LUMO may have a different spin against to HOMO. It is worth noting that spin polarization is realized in the higher charged states. To analyze each spin- polarized state, I will compare DOSs of the charged clusters with each other in Figs. 2.10(b)-(f). As in DOS of the neutral cluster (Fig. 2.10(a)), the red and blue curves of the Au(5d) and Au(5d)-S(3p) regimes are almost symmetrical indicating that the up and down spins form a pair. In contrast, DOSs associated with the Au₇ core depend on the charged states and particularly DOSs for the up and down spins in the charged states of 3−, 2−, and 3+ are asymmetrical. These spectral patterns show that the spin polarization is realized in such charged states. Furthermore, the unpaired electrons contributing to these spin-polarized states are localized at the Au₇ core. Figure 2.11 shows the KS orbitals related to the spin-polarized states. This figure schematically demonstrates that the unpaired electrons are localized at the core but not distributed around the Au-S cage.

Table 2.3: The HOMO-LUMO gaps and the spin multiplicities of Au₂₅(SCH₃)₁₈ in different charged states. The spin multiplicities are given in terms of (2S + 1) with S being the total spin quantum number.

charge 3− 2− 1− 0 1+ 3+

HOMO-LUMO gap (eV) 0.6 0.5 0.7 0.4 2.2 0.8

2S + 1 3 4 1 2 1 3

HOMO

HOMO-1

3- 2- 0 3+

Figure 2.11: The top views of the KS orbitals associated with the unpaired electron. The orbitals are either HOMO or HOMO-1. The orbitals for n = 3- (HOMO-1), n = 2- (HOMO-1), and n = 3+ (HOMO) are doubly degenerate.

To illustrate the magnetic properties of the gold-thiolate cluster more clearly, I show in Figure 2.12 contour plots of the spin densities of the different charged states of 3−, 2−, 0, and 3+. The spin densities in the figure are given as difference between the up (red) and down (blue) electron spin densities. All the up spin densities (red) associated with the Au(6s) orbital are localized in the outer part of the Au7core. Very small (almost negligible) spin densities (ρ = -0.004e) mainly consisting of the down spin distribute around the Au₇ core. The contour plots for the charged states of 3-, 2-, and 3+ clearly show that the inhomogeneous spin distributions are localized in the Au₇ core. In particular, in the charged state of 2-, the up spin densities associated with the Au(6s) orbital distribute around the Au₇ core ring whereas the down spin densities are localized at the core. In contrast, in the contour plot of the neutral state, the spin density distribution becomes very subtle. All these spin density maps completely reflect the results described above, that is, the high-spin polarizability is realized in the charged state of 2- and the corresponding inhomogeneous spin densities distribute in the Au₇ core.

Following the explanations by Vager and Naaman, and Hernando et al., the key in- gredient to understand the magnetism is the giant magnetic moments. Such magnetic moments can be induced under the conditions that spin-polarized electrons are local- ized and magnetic anisotropy exists. These conditions are also satisfied in the present gold-thiolate cluster. The spin-polarized electrons are localized at the Au₇ core and the anisotropy is expected because the Au₇ core has an almost planar structure. Fur- thermore, since the Au7 core is enclosed by the robust Au-S cage, the spin-polarized

3- 2-

0 3+

up down

ρ = 0.012e 0.008e 0.004e

ρ = −0.012e −0.008e −0.004e

Figure 2.12: The top views of contour plots of the spin densities given as difference between the up and down electron spin densities. The charged states are 3-, 2-, 0, and 3+, respectively.

electrons will still be localized when the gold-thiolate cluster condenses to form larger compounds. The charged states of the gold-thiolate cluster are changed by introducing positive or negative counter ions. Therefore, the gold-thiolate compounds with unusual magnetism can be constructed by building up the gold-thiolate cluster Au₂₅(SCH₃)₁₈ through controlling the charged states.

Absorption spectra

Figure 2.13shows absorption spectra of 1, 2, and 3 to the left and the corresponding experimental spectra to the right. The calculated absorption peaks were convoluted by the Lorentz function with a width of 40 nm. The first peak appears at ∼ 680 nm, irrespective of the charged state, whereas there is a glimpse of peaks around 800 nm both in the spectra of 2 and 3. The spectral patterns are in good accord with the experimental one. Peak patterns of 1 and 3 are relatively similar, and of 3 shows peak splitting around 680 nm. To analyze these spectral patterns, KS-orbitals concerning

Intensity (Arb. units)

Wave length (nm) EXP. Calc.

1- 2- 3-

0

1+ 2+

700

500 900

700

500 900

Figure 2.13: Absorption spectra of Au₂₅(SCH₃)₁₈of calculated (left) and experimental (right) in the charged states are shown.

to the first peaks will be analyzed below.

Although, KS-orbital is just a mathematical tool and has little physical meanings, it is not so bad to use them for qualitative interpretation of molecular orbitals if we restrict ourselves to the discussion near the Fermi level [154]. To confirm this, I have performed HF/TZVP calculation on 1+ cationic state in closed shell model and com- pared the result with that of B3LYP/TZVP. Moreover, the MP2 calculation following to the HF/TZVP calculation shows the largest changes in occupation numbers of 5.79

%. This suggests that the HF occupations are acceptable in this gold-thiolate sys- tem and the comparison between HF and DFT calculations can make a sense. From the comparisons, I have obtained that the orbital orders around the Fermi levels are the same and the orbital shapes are indistinguishable. The HOMO-LUMO gaps and energy differences between HOMO(= HOMO-1) and HOMO-2 in HF calculation are larger than B3LYP one. Thus, the KS-orbital analyses on the absorption spectra provide good qualitative and intuitive pictures.

Figure 2.14 shows the KS orbital diagrams concerning to the first peaks. Some representative electronic transitions associated with the absorption peaks in Fig.2.13 are assigned. All the charged states have the same KS orbital characters with minor relaxations thorough varying the charges as reported experimentally [5, 91]. In 2 and 3, the frontier electrons occupy LUMO of 1 which is doubly degenerated in nature. While such occupation breaks the closed shell structure of 1, the electronic structure

748nm 689nm

1+ 0 1-

787 ~ 827nm 727 ~ 685nm

832nm 690nm

Figure 2.14: HOMO-LUMO

of 3 has quasi-closed shell character. The electronic transitions of the first peak of 1 is attributed to the HOMO-LUMO transitions, both HOMO and LUMO are composed mainly of Au₇. In the spectra of 2 and 3, the first peaks are attributed to the transition from KS orbitals of Au₇ to Au(6s) of the Au-S cages, and the small peaks around 800 nm correspond to the first peak of 1. The peak splitting observed around 680 nm in 2 can be explained by the disordered electronic structure compared to the other charged states, but having minor effects on the spectral curve. Thus, it can be concluded that the similarity in the peak positions between the spectra of 1, and 2 and 3 are just a coincidence, which arise from the similarity of the relative spacing of the energy levels concerned. In contrast, the similarity between 2- and 3- spectra results from the same nature in the electronic configurations and transitions.

2.1.4 Short summary

I have presented the DFT study of the model cluster [Au25(SCH3)18]⁺ mimicking the extraordinarily stable glutathione(GSH)-protected gold cluster Au₂₅(SG)₁₈ ex- perimentally observed very recently [1–3]. Three types of optimized structures were derived from the different core clusters of the fcc Au25 and of the vertex-sharing cen- tered icosahedral Au₁₃ dimer. The most preferred optimized structure, FCC2, which is based on the truncated fcc Au₂₅ cluster, shows the unusually large HOMO-LUMO gap and sufficiently reproduces the experimental data of the XRD and absorption spectra. I have found that FCC2 can be partitioned into three types of subsystems, the Au₇ core cluster, the Au₁₂(SCH₃)₁₂ complex-like ring, and the two Au₃(SCH₃)₃ rings. This classification leads to a novel structural picture that the bare gold cluster is enclosed by the -Au-S- repeated network. Such a structural understanding reveals the physicochemical properties of small thiolated gold clusters in a new and different way.

The unrestricted DFT calculations have been performed to investigate the spin- polarized electronic structures of the core-in-cage geometry, Au7(AuSCH3)12[(AuSCH3)3]2,

with different charged states from 3− to 3+. While the spin multiplicity (2S +1) of the neutral cluster is 2, the higher-spin polarization is realized in the charged states. Fur- thermore, the unpaired electrons associated with these high spin-polarized states are always localized at the Au7 core that is encapsulated by the robust Au-S cage. There- fore, we can control the degree of spin polarization by changing the charged states of the gold-thiolate cluster by introducing counter ions. The cluster will induce magnetic anisotropy due to the planar geometry of the Au₇ core even when integrated. Such controllable magnetic moments localized at the core cluster can be utilized to develop single molecule magnets and also magnetic storages by building up the gold-thiolate clusters. Optical absorption spectra of the charged states reproduce the experimental spectral curve in a good manner. Analysis of the absorption spectra with the elec- tronic structure classifies the similarity of the peak position between cationic and the other states a coincidence. On the other hand, the anionic and neutral state have same electronic configurations leading the similar absorption spectral patterns having same peak positions and shoulders over the first peak. From the total energy differ- ences, IP, EA, and VDE are estimated, from which I conclude that the anionic state can also be observed in a gas phase. Beyond the structural motif discovered at large gold-thiolate cluster, the cluster around 1 nm shows distinct geometrical feature and electronic properties and this suggests a need for more detailed analysis by experi- mental and theoretical works. These fabricated gold-thiolate clusters open up the new class of functional cluster material science.

2.2 Au

oligomeric clusters

Tsukuda and coworkers made significant progress toward realizing the cluster-assembled compounds in a bottom-up approach [6]. They synthesized a gold cluster compound [Au₂₅(PPh₃)₁₀(SC₂H₅)₅Cl₂]²⁺ (1) and characterized its geometric structure through single X-ray crystal analysis. The cluster has a unique structure in the sense that the Au₂₅core is constructed by bridging two icosahedral Au₁₃clusters with thiolates shar- ing a vertex gold atom and is terminated by two chlorine atoms. This biicosahedral structure is conceptually close to the clusters of clusters by Teo et al., and thus it is considered to be a dimer consisting of the two icosahedral Au₁₃units. The absorption spectrum showed that such dimerization gives rise to a new electronic level retaining the electronic properties of the individual Au13 constituents. Therefore, the icosahe- dral Au₁₃ cluster incorporating with thiolates will be a typical prototype of building blocks of assembled gold clusters.

In this section, characterization of geometric and electronic structures of a dimer cluster [Au₂₅(PH₃)₁₀(SCH₃)₅Cl₂]²⁺ (2) mimicking the cluster 1 will be discussed. Then, the mechanism of the dimerization will be studied theoretically. I will further

discuss polymerization, as an example, the trimerized gold cluster [Au₃₇(PH₃)₁₀(SCH₃)₁₀Cl₂]⁺.

2.2.1 Method of computation

Density functional theory (DFT) calculations have been carried out for the gold cluster 2. In this model cluster, the triphenylphosphines of 1 were replaced with the phosphines, and the alkanethiolates were replaced with the methanethiolates. I have adopted such a simplification of the ligands frequently used in previous calcula- tions [135,137,139,140,155,156]. Geometry optimization of the cluster 2 was performed starting from the initial guess structure taken from the single-crystal X-ray data of 1. This initial structure is closed to C_5h molecular symmetry. I did not assume such a high molecular symmetry but performed the geometry optimization of the cluster within C_s molecular symmetry.

I should refer to the accuracy of the present DFT calculations with the B3LYP functional. The determination of the most preferable functional, particularly for het- erogeneous clusters such as the present metal-molecule compounds, is beyond the scope of the present work. I have confirmed that at least for the present gold cluster compounds, the calculations with B3LYP compared with the PBE functional [157] reasonably reproduced the experimental results. Although the optimized structure based on the calculation with PBE is in slightly better accord with the experimental result, the calculation failed to reproduce the detailed absorption spectral patterns of the experiment. For references of comparison of functionals in gold-thiolate compound systems, see [139] and the following papers [153, 158]